COMMON FIXED POINT THEOREM FOR GENERALIZED T-HARDY- …

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*Corresponding author

Received September 24, 2013

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COMMON FIXED POINT THEOREM FOR GENERALIZED T-HARDY-

ROGERS CONTRACTION MAPPING IN A CONE METRIC SPACE

A.K. DUBEY1,*

, RITA SHUKLA2, R.P. DUBEY

3

1Department of Mathematics, Bhilai Institute of Technology, Bhilai House, Durg 491001, Chhattisgarh, INDIA

2Department of Mathematics, Shri Shankracharya College of Engineering and Technology (SSGI), Bhilai, Durg

490020, Chhattisgarh, INDIA

3Department of Mathematics, Dr. C.V.Raman University, Bilaspur, Chhattisgarh, INDIA

Copyright © 2014 A.K. Dubey et. al. This is an open access article distributed under the Creative Commons Attribution License, which

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract: In the present paper we improve and generalize common fixed point theorem for T-Hardy-Rogers

contraction mapping in the setting of cone metric space.

Keywords: Cone metric space, Banach operator pair, T-contraction, Hardy-Rogers type contraction, common

fixed point.

2000 AMS Subject Classification: 47H10, 54H25.

1. Introduction

It is well known that the classic contraction mapping principle of Banach is a fundamental

result in fixed point theory. Several authors have obtained various extensions and

generalizations of Banach’s theorem by considering contractive mappings on many different

metric spaces.

Available online at http://scik.org

Adv. Inequal. Appl. 2014, 2014:18

ISSN: 2050-7461

2 A.K.DUBEY, RITA SHUKLA, AND R.P.DUBEY

In 1977, Rhoades [16] considered 250 types of contractive definitions and analyzed the

relationship among them. In 2009, A Beiranvand et al. [1] introduced new classes of

contractive functions-T-contraction and T-contractive mappings and then they established

and extended the Banach contraction principle and the Edelstein’s fixed point theorems.

Recently, Huang and Zhang [2] introduce the notion of cone metric space. He replaced real

number system by ordered Banach space. He also gave the condition in the setting of cone

metric spaces. These authors also described the convergence of sequences in the cone metric

spaces and introduce the corresponding notion of completeness. Subsequently, many authors

have generalized the results of Huang and Zhang and have studied fixed point theorems for

different types of cones, see for instance [5], [6], [9], [10], [11], [14], [19] and [20].

In sequel, J.R. Morales and E. Rojas [12], [13] obtained sufficient condition for the existence

of a unique fixed point of T-Kannan contractive, T-Zamfirescu, T-contractive mappings etc,

on complete cone metric spaces. Afterwards; in [3] R. Sumitra et al. have proved common

fixed point theorem for a Banach pair of mappings satisfying T-Hardy-Rogers type

contraction condition in cone metric spaces. In the sequel, we need a definition which was

introduced and called Banach operator of type by Subrahmanyam [17]. Recently Chen and

Li [7] extended the concept of Banach operator of type to Banach operator pair and proved

various best approximation results using common fixed point theorems for nonexpansive

mappings.

The aim of this paper is to prove common fixed point theorem for generalized T-Hardy-

Rogers contraction mappings in the setting of cone metric spaces. Our result is generalization

of the result [3].

2. Preliminary Notes

First, we recall some standard notations and definitions which we need them in the sequel.

Definition 2.1([2]) .Let be a real Banach space and a subset of . is called a cone if

and only if

(i) is closed, non-empty and ,

COMMON FIXED POINT THEOREM 3

(ii) for all and non-negative real numbers

(iii) and .

Given a cone a partial ordering is defined as on with respect to by if and

only if . It is denoted as will stand for denotes the

interior of . The cone is called normal if there is a number such that for all

implies (2.1)

The least positive number satisfying the above is called the normal constant of

Definition 2.2([2]) .Let be a non-empty set. Suppose is a real Banach space, is a cone

with and is partial ordering with respect to If the mapping

satisfies,

(i) for all and if and only if ;

(ii) for all

(iii) for all

Then is called a cone metric on and is called cone metric space .


Example 2.3([2]). Let and

such that where is a constant.Then is a cone

metric space.

Lemma 2.4([2]). Let be a cone metric space and be a normal cone with normal

constant A sequence in converges to if and only if as .

Lemma 2.5([2]). Let be a cone metric space and be a normal cone with normal

constant A sequence in is a Cauchy sequence if and only if as

.

Definition 2.6[12]. Let be a cone metric space and be a sequence in . Then,

(i) converges to if for every with , there is the set of all

natural numbers such that for all ,

.

It is denoted by or

(ii) If for every there is a number such that for all

then is called a Cauchy sequence in

(iii) is called a complete cone metric space, if every Cauchy sequence in is

convergent


(iv) A self mapping is said to be continuous at a point if

implies that for every in

Definition 2.7. A self mapping of a metric space is said to be a contraction

mapping, if there exists a real number such that for all

(2.2)

The following definition is given by Beiranvand et al. [1].

Definition 2.8([1]). Let and be two self-mappings of a metric space The self

mapping of is said to be T-contraction, if there exists a real number such

that

(2.3)

for all

If the identity mapping, then the Definition 2.8 reduces to Banach contraction

mapping.

The following example shows that a T-contraction mapping need not be a contraction

mapping.

Example 2.9. Let be with the usual metric. Let define two mappings

as


It is clear that, is not contraction but is T-contraction, since,

Definition 2.10([1]). Let be a self mapping of a metric space Then

i) A mapping is said to be sequentially convergent, if the sequence in X is

convergent whenever is convergent.

ii) A mapping is said to be sub-sequentially convergent, if has a convergent

subsequence whenever is convergent.

Definition 2.11([17]). Let be a self mapping of a normed space . Then is called a

Banach operator of type if

for some and for all

This concept was introduced by Subrahmanyam [17], then Chen and Li [7] extended this

as following:


Definition 2.12([7]).Let and be two self mappings of a non-empty subset of a

normed linear space . Then is a Banach operator pair, if any one of the following

conditions is satisfied:

(i) i.e. is T-invariant.

(ii) for each .

(iii) for each .

(iv) for some

The following corollary of Rezapour [15] will be needed in the sequel.

Corollary 2.13([15]).Let a,b,c,u the real Banach space.

(i) If and b then

(ii) If and b then

(iii)If for each then

Remark 2.14([10]). If 0 and then there exists such that for all

it follows that

3. Main Result


Theorem 3.1.Let and be three continuous self mappings of a complete cone metric

space . Assume that is an injective mapping and is a normal cone with normal

constant. If the mappings and satisfy

(3.1)

for all where are all non negative constants such that

then and have a unique common fixed point in

Moreover, if and are Banach pairs, then and have a unique common fixed

point in

Proof. Let as an arbitrary element and define the sequences in

such that

and for each . Consider,


+ (3.2)

Next consider,

(3.3)

Adding inequalities (3.2) and (3.3),


where

Proceeding further,

(3.4)

Next, to claim that is a Cauchy sequence. Consider such that

).

From (2.1), it follows that

‖

d( (3.5)


Since as . Therefore ‖d( as Thus

is a Cauchy sequence in As is a complete cone metric space, there exists

such that

Since is sub-sequentially convergent, has a convergent sub-sequence such that

As is continuous

(3.6)

By the uniqueness of the limit, .

Since is continuous Again as is continuous,

Therefore

(3.7)

Now consider,


Therefore,

(3.8)

Let be arbitrary. By (3.6),

Similarly by (3.7), it follows that

.

Then, (3.8) becomes


.

Thus for each Now, using Corollary 2.13 (iii), it follows that

which implies that . As is injective, Thus is the fixed point of

Similarly, it can be established that, is also the fixed point of . That means is common

fixed point of and

To prove uniqueness: If is another common fixed point of and , then


a contradiction. Hence which implies As is injective, is

the unique common fixed point of and

Since we have assumed that and are Banach pairs; and commutes at

the fixed point of and respectively. This implies that for . So

which gives that is another fixed point of It is true for too. By the

uniqueness of fixed point of Hence is the unique common

fixed point of and in .

Conflict of Interests

The authors declare that there is no conflict of interests.

REFERENCES

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COMMON FIXED POINT THEOREM FOR GENERALIZED T-HARDY- …